openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /HierarchyGraph.lean
| import OAI.Computability.DegreeRigidity.Constructibility.UniformDefSuccessor | |
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleStages | |
| import OAI.Computability.DegreeRigidity.Syntax.CheckGraphConstruction | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| def StageStep (s f r x A : ZFSet.{u}) : Prop := | |
| s ⊆ A ∧ | |
| (∀ z ∈ A, z ∈ s ∨ ∃ y ∈ x, ∃ w ∈ r, ZFSet.pair y w ∈ f ∧ z ∈ w) ∧ | |
| ∀ y ∈ x, ∀ w ∈ r, ZFSet.pair y w ∈ f → w ⊆ A | |
| def HierarchyGraph (M s d r f : ZFSet.{u}) : Prop := | |
| Transitive d ∧ FunctionGraph d r f ∧ | |
| ∀ x ∈ d, ∀ v ∈ r, ZFSet.pair x v ∈ f → | |
| ∃ A ∈ M, StageStep s f r x A ∧ v = definablePower A | |
| theorem HierarchyGraph.correct {M R d r f : ZFSet.{u}} | |
| (h : HierarchyGraph M (seed R) d r f) (x : ZFSet.{u}) (hx : x ∈ d) | |
| (v : ZFSet.{u}) (hv : v ∈ r) (hfv : ZFSet.pair x v ∈ f) : | |
| v = definablePower (stage R x) := by | |
| induction x using ZFSet.inductionOn generalizing v with | |
| | h x ih => | |
| obtain ⟨A,_,hs,rfl⟩ := h.2.2 x hx v hv hfv | |
| congr 1 | |
| apply ZFSet.ext; intro z | |
| rw [mem_stage] | |
| constructor | |
| · intro hz | |
| rcases hs.2.1 z hz with hz|⟨y,hy,w,hw,hfw,hz⟩ | |
| · exact Or.inl hz | |
| · rw [ih y hy (h.1 x hx y hy) w hw hfw] at hz | |
| exact Or.inr ⟨y,hy,hz⟩ | |
| · rintro (hz|⟨y,hy,hz⟩) | |
| · exact hs.1 hz | |
| · obtain ⟨w,hw,hfw,_⟩ := h.2.1.2 y (h.1 x hx y hy) | |
| rw [← ih y hy (h.1 x hx y hy) w hw hfw] at hz | |
| exact hs.2.2 y hy w hw hfw hz | |
| theorem HierarchyGraph.mem_iff {M R d r f : ZFSet.{u}} | |
| (h : HierarchyGraph M (seed R) d r f) (z : ZFSet.{u}) : | |
| z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (definablePower (stage R x)) := by | |
| constructor | |
| · intro hz | |
| obtain ⟨x,hx,v,hv,rfl⟩ := h.2.1.1 z hz | |
| exact ⟨x,hx,by rw [h.correct x hx v hv hz]⟩ | |
| · rintro ⟨x,hx,rfl⟩ | |
| obtain ⟨v,hv,hfv,_⟩ := h.2.1.2 x hx | |
| rw [← h.correct x hx v hv hfv] | |
| exact hfv | |
| theorem HierarchyGraph.unique {M N R d r t f g : ZFSet.{u}} | |
| (hf : HierarchyGraph M (seed R) d r f) (hg : HierarchyGraph N (seed R) d t g) : f = g := by | |
| apply ZFSet.ext; intro z | |
| exact (hf.mem_iff z).trans (hg.mem_iff z).symm | |
| noncomputable def stageImage (s x f : ZFSet.{u}) : ZFSet.{u} := | |
| s ∪ (ZFSet.sUnion (iterUnion 2 f)).sep | |
| (fun z => ∃ y ∈ x, ∃ w ∈ iterUnion 2 f, ZFSet.pair y w ∈ f ∧ z ∈ w) | |
| theorem mem_stageImage (s x f z : ZFSet.{u}) : | |
| z ∈ stageImage s x f ↔ z ∈ s ∨ ∃ y ∈ x, ∃ w, ZFSet.pair y w ∈ f ∧ z ∈ w := by | |
| rw [stageImage,ZFSet.mem_union,ZFSet.mem_sep] | |
| apply or_congr_right | |
| constructor | |
| · rintro ⟨_,y,hy,w,_,hf,hz⟩; exact ⟨y,hy,w,hf,hz⟩ | |
| · rintro ⟨y,hy,w,hf,hz⟩ | |
| have hw := second_mem_doubleUnion hf | |
| exact ⟨ZFSet.mem_sUnion.mpr ⟨w,hw,hz⟩,y,hy,w,hw,hf,hz⟩ | |
| theorem stageImage_mem (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) | |
| {s x f : ZFSet.{u}} (hs : s ∈ M) (hx : x ∈ M) (hf : f ∈ M) : stageImage s x f ∈ M := by | |
| let C : Context M := ⟨hM,hT.pairing,hT.union,hT.powerSet,hT.separation.finitePrefix.bounded,hT.infinity⟩ | |
| have hr := iterUnion_mem M hM hT.union hf 2 | |
| have hd := (((defPairMem C 1 0 4).and (member_definable C 2 0)).existsMem 4).existsMem 1 | |
| let e := cons x (cons f (fun _ => iterUnion 2 f)) | |
| have he : ∀ i, e i ∈ M := by intro i; rcases i with _|_|i; exact hx; exact hf; exact hr | |
| exact binary_union_mem M hM hT.pairing hT.union hs | |
| (hd.sep_mem C e he (union_mem M hM hT.union hr)) | |
| theorem hierarchyGraph_of_mem_iff (M R d f : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) | |
| (hs : seed R ∈ M) (hdM : d ∈ M) (hfM : f ∈ M) (hd : Transitive d) | |
| (hf : ∀ z, z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (definablePower (stage R x))) : | |
| HierarchyGraph M (seed R) d (iterUnion 2 f) f := by | |
| have hpair (x v : ZFSet.{u}) : ZFSet.pair x v ∈ f ↔ x ∈ d ∧ v = definablePower (stage R x) := by | |
| rw [hf] | |
| constructor | |
| · rintro ⟨y,hy,he⟩ | |
| obtain ⟨rfl,hv⟩ := ZFSet.pair_inj.mp he | |
| exact ⟨hy,hv⟩ | |
| · rintro ⟨hx,rfl⟩; exact ⟨x,hx,rfl⟩ | |
| refine ⟨hd,⟨?_,?_⟩,?_⟩ | |
| · intro z hz | |
| obtain ⟨x,hx,rfl⟩ := (hf z).mp hz | |
| exact ⟨x,hx,_,second_mem_doubleUnion hz,rfl⟩ | |
| · intro x hx | |
| have hp := (hpair x _).mpr ⟨hx,rfl⟩ | |
| exact ⟨_,second_mem_doubleUnion hp,hp,fun v _ hv => (hpair x v).mp hv |>.2⟩ | |
| · intro x hx v _ hv | |
| have he := (hpair x v).mp hv |>.2 | |
| subst v | |
| have hA : stageImage (seed R) x f = stage R x := by | |
| apply ZFSet.ext; intro z | |
| rw [mem_stageImage,mem_stage] | |
| apply or_congr_right | |
| constructor | |
| · rintro ⟨y,hy,w,hw,hz⟩ | |
| rw [(hpair y w).mp hw |>.2] at hz | |
| exact ⟨y,hy,hz⟩ | |
| · rintro ⟨y,hy,hz⟩ | |
| exact ⟨y,hy,_,(hpair y _).mpr ⟨hd x hx y hy,rfl⟩,hz⟩ | |
| refine ⟨stage R x,hA ▸ stageImage_mem M hM hT hs (hM d hdM x hx) hfM,?_,rfl⟩ | |
| refine ⟨seed_subset_stage R x,?_,?_⟩ | |
| · intro z hz | |
| rcases (mem_stage R x z).mp hz with hz|⟨y,hy,hz⟩ | |
| · exact Or.inl hz | |
| · have hp := (hpair y _).mpr ⟨hd x hx y hy,rfl⟩ | |
| exact Or.inr ⟨y,hy,_,second_mem_doubleUnion hp,hp,hz⟩ | |
| · intro y hy w _ hw | |
| rw [(hpair y w).mp hw |>.2] | |
| exact definablePower_subset_stage R hy | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |